Ctex ontology job

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% This is the title of the article
\ Title {owl job}

% This is the author of the article
\ Author {Xu Weihong}

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\ Newpage
\ Section {Exercise 1}
\ Subsection {question}
Exercise 1. Describe-both verbally and formally-the extension of the following
Concepts with respect to the interpretation $ \ mathcal {I} $ defined in Example 16:
\ Begin {enumerate} [(a)]
\ Item $ \ forall hassuccessor ^ {-}. Positive $
\ Item $ \ exists multipleof. Self $
\ Item $ \ exists multipleof. \ exists hassuccessor ^ {-}. \ exists hassuccessor ^ {-}. \ {zero \} $
\ Item $ \ geq 10lessthan ^ {-}. Prime $
\ Item $ \ neg prime \ sqcap \ Leq 2multipleof. \ top $
\ Item $ \ exists lessthan. Prime $
\ Item $ \ forall multipleof. (\ exists hassuccessor -. \ {zero \} \ sqcup \ exists multipleof. \ exists has successor ^ {-}. \ exists hassuccessor ^ {-}. \ {zero \}) $
\ Begin {enumerate} [-]
\ Item $ N _ {\ mathcal I }=\{ zero \} $
\ Item $ N_C =\{ Prime, positive \} $
\ Item $ n_r =\{ hassuccessor, lessthan, multipleof \} $
\ End {enumerate}
\ End {enumerate}
Now, we define $ {\ mathcal I} $ as follows: let $ \ Delta ^ \ mathcal {I }={\ mathcal n }=\{ 0, 1, 2 ,... \} $, I. E ., the set of all natural numbers including zero. furthermore, we let $ zero ^ {\ mathcal I} = 0 $, as well as $ prime ^ {\ mathcal I }=\{ n | n is a prime number \} $
And $ positive ^ {\ mathcal I }=\{ n | n> 0 \} $. For the roles, we define
\ Begin {enumerate} [-]
\ Item $ hassuccessor ^ {\ mathcal I }={< N, N + 1> | n \ in {\ mathcal n }\} $
\ Item $ lessthan ^ {\ mathcal I }={< N, N'> | n <n', N, N' \ in {\ mathcal n }\} $
\ Item $ multipleof ^ {\ mathcal I }={< N, N'> | \ exists k, n = K, n', N, N ', k \ in {\ mathcal n }\} $
\ End {enumerate}
\ Subsection {solution}

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\ Section {Exercise 2}
\ Subsection {question}
Decide whether the following axioms are satisfied by the Interpretation $ {\ mathcal I} $ from Example 16.
\ Begin {enumerate} [(a)]
\ Item $ hassuccessor \ sqsubseteq lessthan $
\ Item $ \ exists hassuccessor ^ {-}. \ exists hassuccessor ^ {-}. \ {zero \} \ sqsubseteq prime $
\ Item $ \ top \ sqsubseteq \ forall multipleof ^ {-}. \ {zero \} $
\ Item $ DIS (divisileby, lessthan ^ {-}) $
\ Item $ multipleof \ CIRC multipleof \ sqsubseteq multipleof $
\ Item $ \ top \ sqsubseteq \ Leq 1hassuccessor. Positive $
\ Item $ zero \ not \ approx zero $
\ Item $ \ Leq 1mutipleof ^ {-}. \ top (zero) $
\ Item $ \ top \ sqsubseteq \ forall lessthan. \ exists lessthan. (prime \ sqcap \ exists hassuccessor. \ exists hassuccessor. Prime) $
\ End {enumerate}
\ Subsection {solution}
\ Newpage
\ Section {exercise 11}
\ Subsection {question}
Show that the following equivalences are not valid:
\ Begin {enumerate} [(a)]
\ Item $ \ exists \ textbf {r}. (C \ sqcap d) \ equiv \ exists \ textbf {r}. c \ sqcap \ exists \ textbf {r}. d $
\ Item $ C \ sqcap (d \ sqcup e) \ equiv (C \ sqcap d) \ sqcup e $
\ Item $ \ exists \ textbf {r }. \ {\ textbf {A }\}\ sqcap \ exists \ textbf {r }. \ {\ textbf {B }\}\ equiv \ geq 2. \{\ textbf {A}, \ textbf {B }\}$
\ Item $ \ exists \ textbf {r }. \ top \ sqcap \ exists \ textbf {s }. \ top \ equiv \ exists \ textbf {r }. \ exists \ textbf {r }^ {-}. \ exists \ textbf {s }. \ top. $
\ End {enumerate}
\ Subsection {solution}

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\ Section {summary}
\ Subsection {description logics description logic}
All DLS are based on a vocabulary [Signature] containing individual names [constants], concept Names [unary predicates] and role Names [binary predicates]. two specific class names, $ \ top and \ bot $, denote the concept containing all individuals and the empty
Concept, respectively.

Usually, a DL knowledge base [theory] is partitioned into an assertional part, called abox and a terminological part, which is further subdivided int tbox and rbox.
\ Newpage
\ Subsubsection {introduction}
\ Begin {enumerate} [-]
\ Item abox: The abox contains assertional knowledge [Ground facts], the notation of which coincides with fol: there are \ textsl {\ textbf {concept assertions} And \ textsl {\ textbf {role assertions }}
\ Begin {enumerate} [$ \ CIRC $]
\ Item concept assertions: C ()
\ Item role assertions: R (A, B)
\ End {enumerate}
\ Item tbox: tbox contains \ textsl {\ textbf {universal statements}. The notation used in DLS does not need variables and is already red by set theory.
\ Begin {itemize}
\ Item we can specify \ textbf {subsumptions}: $ C \ sqsubseteq d $
\ Item a specific feature of DLS is that concept names can be combined into complex concepts by \ textbf {boolean operators }:$ \ exists \ textbf {r}. C $
\ Item \ textbf {role inverses} can be used to "traverse" roles backeward: $ \ exists \ textbf {r }. \ top \ sqsubseteq \ forall \ textbf {r} ^ {-}. C $
\ Item \ textbf {cardinality constraints} allow for specifying the number of related instances: $ \ geq 2. Married. \ top $
\ Item by means of \ textbf {nominals}, classes can be defined by enumerating their instances: $ \ exists R. C \ sqsubseteq \ {... \} $
\ End {itemize}
\ Item rbox: The rbox of DL knowledge base allows for further, role-centric Modeling Features
\ Begin {enumerate}
\ Item \ textbf {role customization sion} statements: $ \ textbf {r} \ sqsubseteq \ textbf {r} $
\ Item a more general and expressive variant of role comprehensions are \ textbf {role-chain} axioms: $ R_1 \ CIRC R_2... \ CIRC R_N \ sqsubseteq r$
\ End {enumerate}
\ End {enumerate}
\ Newpage
\ Subsubsection {Syntax of description logics}
Most of today's mainstream DLS are, in fact, sublanguages of $ {\ mathcal srpiq} $
\ Begin {enumerate} [-]
\ Item the set $ n_ I $ of individual names contains all names used to denote singular entity in our domain of interest.
\ Item the set $ N-C $ of concept names contains names that refer to types, categories, or classes of entities, usually characterized by common properties.
\ Item the set $ n_r $ of role names contains nameds that denote binary relationships which may hold between individuals of a domain.
\ End {enumerate}
Having these name sets at hand, we can now turn to the three building blocks of $ {\ mathcal sroiq} $ knowledge bases: rbox, tbox and abox.
\ Begin {enumerate}
\ Newpage
\ Item rbox \ {roles | roles. dependencies \}

A role can be
\ Begin {itemize}
\ Item a role name \ textbf {r}
\ Item an inverted role name \ textbf {$ R ^ {-} $}
\ Item the universal role \ textbf {u}
\ End {itemize}

A \ textsl {role visibility sion axiom} (RIA) is a statement of the Form $ R_1 \ CIRC R_2 \ CIRC... \ CIRC R_N \ sqsubseteq r$ $ where $ r_1, r_2 ,..., r_N, r$ are roles
\ Begin {itemize}
\ Item given a set of RIAs, roles are divided into \ textsl {\ textbf {simple }}and \ textsl {\ textbf {non-simple} roles.
\ Item roughly, role are non-simple if they may occur on the RHS of a complex Ria.
\ Item more precisely,
\ Begin {itemize}
\ Item for any RIA \ textbf {$ R_1 \ CIRC R_2 \ CIRC... \ CIRC R_N \ sqsubseteq r$} with $ n> 1 $, \ textbf {r} is non-simple
\ Item any RIA $ s \ sqsubseteq r$ with \ textbf {s} non-simple, \ textbf {r} is non-simple
\ Item all other properties are simple
\ End {itemize}
\ End {itemize}

A role disjointness statement has the form $ DIS (S1, S2) $ where S1 and S2 are simple roles.

\ Begin {tabular} {| c |}
\ Hline
% After \: \ hline or \ Cline {col1-col2} \ Cline {col3-col4 }...
Concept Expressions & meaning \\
\ Hline
$ \ Top $ and $ \ bot $ & concept expressions \\
$ A_1,..., a_n $ & individual names \\
$ \ {A_1,..., a_n \} $ & concept expression \\
$ \ Lnot C $ and $ C \ sqcap d $ and $ C \ sqcup d $ & concept expressions \\
$ \ Exists R. C $ and $ \ forall R. C $ & concept expressions \\
$ \ Exists R. Self $ and $ \ Leq ns. C $ and $ \ geq ns. C $ & concept expressions \\
\ Hline
\ End {tabular}

For C and D concept expressions, for r a role, for s a simple role
\ Newpage
\ Item tbox \ {concepts | concepts. taxonomic dependencies \}
A general concept compression Sion (GCI) has the form $ C \ sqsubseteq d $ where C and D are concept expressions.

A tbox consists of a set of GCIs.

\ Newpage
\ Item abox \ {individuals | individuals. Concept role. memberships \}

\ Begin {tabular} {| c |}
\ Hline
% After \: \ hline or \ Cline {col1-col2} \ Cline {col3-col4 }...
Abox & meaning \\
\ Hline
C (a) & concept assertion \\
R (A, B) & role assertion \\
$ \ Lnot R (A, B) $ & negated role assertion \\
$ A \ approx B $ & policity statement \\
$ A \ not \ approx B $ & Inequality statement \\
\ Hline
\ End {tabular}
\ End {enumerate}
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\ End {document}

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